Analytical Derivation of Complex Cell Properties from the Slowness Principle

Prof. Dr. Laurenz Wiskott

Sprekeler, Henning

Collaborator: Henning Sprekeler

In the project SFA yields complex cell properties we have shown that SFA applied to quasi-natural image sequences yields quadratic functions that share many properties with physiological complex cells in primary visual cortex. Interestingly, the results do not depend on the natural images. If one uses colored noise images, which look like clouds in the sky, one gets the same results. This means that the conditions under which we get complex cell receptive fields can all be formulated analytically: (i) take colored noise images, (ii) generate little movies from them by rotation, translation, and zoom, (iii) rescale all frames of the movies to 16x16 pixels, (iv) apply slow feature analysis (SFA) with quadratic functions, (v) use the resulting quadratic functions as complex cell receptive fields and analyze them - done. As a consequence, one could hope to also solve the problem analytically.

In order to succeed we had to make two simplifying assumptions. Firstly, since dealing with the borders of the receptive fields turned out to be hard, we assumed that the receptive fields are infinitely large and thus have no borders. Secondly, since we have observed in our simulations that translation is the crucial transformation in shaping the receptive fields, we assumed that translation is infinitely faster than rotation or zoom. With these assumption we were able to derive analytical descriptions of the receptive fields, which reproduce a number of important features of those found in simulations and physiological experiments, see figure.

complex cell receptive fields in physiology, simulations, and theory (101 kB) Figure: A comparison between different complex cell properties in a comparison between physiological experiments, numerical simulations with SFA, and theoretical results.

Publications

2013

Slow Feature Analysis
Wiskott, L.
In D. Jaeger & Jung, R. (Eds.), Encyclopedia of Computational Neuroscience Springer-Verlag Berlin Heidelberg

Publisher's document page

@incollection{Wiskott2013, author = {Wiskott, Laurenz}, title = {Slow Feature Analysis}, booktitle = {Encyclopedia of Computational Neuroscience}, editor = {Jaeger, D and Jung, R}, publisher = {Springer-Verlag Berlin Heidelberg}, year = {2013}, doi = {10.1007/978-1-4614-6675-8_682}, }

Wiskott, L. (2013). Slow Feature Analysis. In D. Jaeger & Jung, R. (Eds.), Encyclopedia of Computational Neuroscience. Springer-Verlag Berlin Heidelberg. http://doi.org/10.1007/978-1-4614-6675-8_682

2011

Slow feature analysis

Wiskott, L., Berkes, P., Franzius, M., Sprekeler, H., & Wilbert, N.

Scholarpedia, 6(4), 5282

@article{WiskottBerkesFranziusEtAl2011,
	author		=	{Wiskott, L. and Berkes, P. and Franzius, M. and Sprekeler, H. and Wilbert, N.},
	title		=	{Slow feature analysis},
	journal		=	{Scholarpedia},
	volume		=	{6},
	number		=	{4},
	pages		=	{5282},
	month		=	{April},
	year		=	{2011},
	doi		=	{10.4249/scholarpedia.5282},
}

A theory of Slow Feature Analysis for transformation-based input signals with an application to complex cells
Sprekeler, H., & Wiskott, L.
Neural Computation, 23(2), 303–335

Journal article page Author's reprint

@article{SprekelerWiskott2011, author = {Sprekeler, Henning and Wiskott, Laurenz}, title = {A theory of Slow Feature Analysis for transformation-based input signals with an application to complex cells}, journal = {Neural Computation}, volume = {23}, number = {2}, pages = {303–335}, year = {2011}, doi = {10.1162/NECO_a_00072}, }

Sprekeler, H., & Wiskott, L.. (2011). A theory of Slow Feature Analysis for transformation-based input signals with an application to complex cells. Neural Computation, 23(2), 303–335. http://doi.org/10.1162/NECO_a_00072

2009

Slowness learning
Sprekeler, H.
Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I

University's eprint server

@phdthesis{Sprekeler2009, author = {Sprekeler, Henning}, title = {Slowness learning}, school = {Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I}, month = {February}, year = {2009}, doi = {10.18452/15897}, }

Sprekeler, H. (2009, February). Slowness learning. Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I. Retrieved from http://edoc.hu-berlin.de/docviews/abstract.php?id=29695

2007

Is slowness a learning principle of the visual system?
Wiskott, L., Franzius, M., Berkes, P., & Sprekeler, H.
In A. Treves, Battaglini, P. P., Chelazzi, L., Diamond, M., & Vallortigara, G. (Eds.), Proc. 39th European Brain and Behaviour Society Meeting (EBBS), Sep 15–19, Triest, Italy (pp. 14–15)

@inproceedings{WiskottFranziusBerkesEtAl2007, author = {Wiskott, Laurenz and Franzius, Mathias and Berkes, Pietro and Sprekeler, Henning}, title = {Is slowness a learning principle of the visual system?}, booktitle = {Proc. 39th European Brain and Behaviour Society Meeting (EBBS), Sep 15–19, Triest, Italy}, editor = {Treves, Alessandro and Battaglini, P. Paolo and Chelazzi, Leonardo and Diamond, Mathew and Vallortigara, Giorgio}, pages = {14–15}, year = {2007}, }

Wiskott, L., Franzius, M., Berkes, P., & Sprekeler, H. (2007). Is slowness a learning principle of the visual system? In A. Treves, Battaglini, P. P., Chelazzi, L., Diamond, M., & Vallortigara, G. (Eds.), Proc. 39th European Brain and Behaviour Society Meeting (EBBS), Sep 15–19, Triest, Italy (pp. 14–15).
Towards an analytical derivation of complex cell receptive field properties
Wiskott, L., Sprekeler, H., & Berkes, P.
In Proc. 7th Göttingen Meeting of the German Neuroscience Society, Mar 29 – Apr 1, Göttingen, Germany (pp. S12–2)

@inproceedings{WiskottSprekelerBerkes2007, author = {Wiskott, Laurenz and Sprekeler, Henning and Berkes, Pietro}, title = {Towards an analytical derivation of complex cell receptive field properties}, booktitle = {Proc. 7th Göttingen Meeting of the German Neuroscience Society, Mar 29 – Apr 1, Göttingen, Germany}, pages = {S12–2}, year = {2007}, }

Wiskott, L., Sprekeler, H., & Berkes, P. (2007). Towards an analytical derivation of complex cell receptive field properties. In Proc. 7th Göttingen Meeting of the German Neuroscience Society, Mar 29 – Apr 1, Göttingen, Germany (pp. S12–2).

2006

Analytical derivation of complex cell properties from the slowness principle
Sprekeler, H., & Wiskott, L.
In Proc. 2nd Bernstein Symposium for Computational Neuroscience, Oct 1–3, Berlin, Germany (p. 67) Bernstein Center for Computational Neuroscience (BCCN) Berlin

@inproceedings{SprekelerWiskott2006d, author = {Sprekeler, Henning and Wiskott, Laurenz}, title = {Analytical derivation of complex cell properties from the slowness principle}, booktitle = {Proc. 2nd Bernstein Symposium for Computational Neuroscience, Oct 1–3, Berlin, Germany}, pages = {67}, publisher = {Bernstein Center for Computational Neuroscience (BCCN) Berlin}, month = {October}, year = {2006}, }

Sprekeler, H., & Wiskott, L.. (2006). Analytical derivation of complex cell properties from the slowness principle. In Proc. 2nd Bernstein Symposium for Computational Neuroscience, Oct 1–3, Berlin, Germany (p. 67). Bernstein Center for Computational Neuroscience (BCCN) Berlin.
Analytical derivation of complex cell properties from the slowness principle
Sprekeler, H., & Wiskott, L.
In Proc. Conference on Mathematical Neuroscience (NEUROMATH 06), Sep 1-4, Andorra (p. 62)

@inproceedings{SprekelerWiskott2006c, author = {Sprekeler, Henning and Wiskott, Laurenz}, title = {Analytical derivation of complex cell properties from the slowness principle}, booktitle = {Proc. Conference on Mathematical Neuroscience (NEUROMATH 06), Sep 1-4, Andorra}, pages = {62}, month = {September}, year = {2006}, }

Sprekeler, H., & Wiskott, L.. (2006). Analytical derivation of complex cell properties from the slowness principle. In Proc. Conference on Mathematical Neuroscience (NEUROMATH 06), Sep 1-4, Andorra (p. 62).
Analytical derivation of complex cell properties from the slowness principle
Sprekeler, H., & Wiskott, L.
In Proc. Berlin Neuroscience Forum, Jun 8–10, Bad Liebenwalde, Germany (pp. 65–66) Berlin: Max-Delbrück-Centrum für Molekulare Medizin (MDC)

@inproceedings{SprekelerWiskott2006, author = {Sprekeler, H. and Wiskott, L.}, title = {Analytical derivation of complex cell properties from the slowness principle}, booktitle = {Proc. Berlin Neuroscience Forum, Jun 8–10, Bad Liebenwalde, Germany}, pages = {65–66}, publisher = {Max-Delbrück-Centrum für Molekulare Medizin (MDC)}, address = {Berlin}, year = {2006}, }

Sprekeler, H., & Wiskott, L.. (2006). Analytical derivation of complex cell properties from the slowness principle. In Proc. Berlin Neuroscience Forum, Jun 8–10, Bad Liebenwalde, Germany (pp. 65–66). Berlin: Max-Delbrück-Centrum für Molekulare Medizin (MDC).
Analytical derivation of complex cell properties from the slowness principle
Sprekeler, H., & Wiskott, L.
In Proc. 15th Annual Computational Neuroscience Meeting (CNS′06), Jul 16–20, Edinburgh, Scotland

@inproceedings{SprekelerWiskott2006b, author = {Sprekeler, Henning and Wiskott, Laurenz}, title = {Analytical derivation of complex cell properties from the slowness principle}, booktitle = {Proc. 15th Annual Computational Neuroscience Meeting (CNS′06), Jul 16–20, Edinburgh, Scotland}, year = {2006}, }

Sprekeler, H., & Wiskott, L.. (2006). Analytical derivation of complex cell properties from the slowness principle. In Proc. 15th Annual Computational Neuroscience Meeting (CNS′06), Jul 16–20, Edinburgh, Scotland.